While the first reaction would be yes, or at least move it occasionally, with trailers it isn't really necessary. Moving it a bit occasionally would be beneficial, but, can't always be done or is very difficult to do, given storage distance for many folks or ease of access or simple convenience (I take the 5th wheel hitch out of the truck when not towing and would have to re-install it just to shift the trailer a foot or two; ain't happening
) The tires may "flat spot", but, with a trailer equipped with ST type tires, the flat spots, due to the different belt configuration will "drive out" when you start towing again. A problem with "taking the weight off" is jacking up the trailer's frame in a manner that doesn't "tweak" the frame (get it out of square). The frame is meant to support the weight in very specific spots and trying to support that weight in other locations can lead to doors that bind, windows that are difficult to open and even leaks caused by twisting stress on seams.
I've never stored a trailer any other way besides parking it where I want it, blocking the wheels and leaving. One thing you can do is add an extra 10 psi to the tires. All tires will bleed down over time and by putting an extra 10 lbs. of air in the tires prior to storage, you shouldn't come back to tires that are low on air. Sitting on low tires (where the sidewall bulges) is more damaging than a full tire just sitting. Make sure the ground that you are parking on is solid or park the trailer on a couple of 2x8 or 2x10 scraps so that the tires don't sit in puddles or sink into the ground and you should be fine.